To determine which of the given formulas could represent a function with a domain of all real numbers, let's analyze each one:

1. $y = \frac{1}{x}$:

   • This is a rational function. The function is undefined when $x = 0$ because division by zero is undefined. Therefore, the domain is all real numbers except $x = 0$.
   • Domain: $\mathbb{R} \setminus \{0\}$.

2. $y = x$:

   • This is a linear function. It is defined for all real values of $x$.
   • Domain: $\mathbb{R}$ (all real numbers).

3. $y = \sqrt{x}$:

   • This is a square root function. The square root is only defined for non-negative values of $x$, meaning $x \geq 0$.
   • Domain: $[0, \infty)$ (all non-negative real numbers).

4. $x = 5$:

   • This represents a vertical line, which is not a function because for a function, each $x$ value should map to exactly one $y$ value. Here, $x = 5$ is a constant and does not represent $y$ as a function of $x$.

Conclusion

The only formula that represents a function with a domain of all real numbers is $y = x$.

Would you like further details on this, or do you have any questions?

Here are 5 related questions to consider:

1. What is the range of the function $y = x$?
2. How do we determine the domain and range of a function?
3. What are the characteristics of a linear function?
4. What happens to the graph of $y = \frac{1}{x}$ near $x = 0$?
5. How can we modify $y = \sqrt{x}$ to make its domain all real numbers?

Tip: When determining the domain of a function, always consider values that might cause the function to be undefined, such as division by zero or taking the square root of a negative number.